vifor is eafily found by infpection. If they are both compound, any common fimple divifor may also be found by infpection. But, when the greateft compound divifor is wanted, the preceding rule is to be applied; only, 2. The fimple divifors of each of the quantities are to be taken out, the remainders in the feveral operations are alfo to be divided by their fimple divifors, and the quantities are always to be ranged according to the powers of the fame letter. The fimple divisors in the given quantities, or in the remainders, do not affect a compound divifor which is wanted; and hence alfo, to make the divifion fucceed, any of the dividends may be multiplied by a' fimple quantity. Befides the fimple divifors in the remainders not being found in the divifors from which they arife, can make no part of the common measure fought; and for the fame reason, if in such a remainder, there be any compound divifor which does not measure the divifor from which it proceeds, it may be taken out. EXAM € If the quantities given are, 8a2b-10ab3 +264, and 9a4b-9a3b2+3a2b3-3ab. The fimple divifors being taken out, viz. 263 out. of the first, it becomes 4a2-5ab+b2, and 3ab out of the fecond, it is 3a-3a2b+ub2· -b3. As the latter is to be divided by the former, it must be multiplied by 4, to make the operation fucceed, and then it is as follows: 4a2-5ab) 12a3—12ab+4ab2-4b3 (za 12a3—15a2b+3ab2 3a2b+ab2-463 This remainder is to be divided by b, and the new dividend multiplied by 3, to make the divifion proceed. Thus, за Ba2+ab-4b2) 12a2—15ab+3b2 (4 12a2+4ab—1662 -19ab+19b2 and this remainder, divided by-196, gives a-b, which being made a divisor, divides 3a2+ab-462, without a remainder, and therefore a-b is the greatest compound divifor; but there is a fimple divifor b, and therefore a-bxb is the greatest common measure required. Prob. II. To reduce a fraction to its loweft terms. Rule. Divide both numerator and denominator by their greatest common measure, which may be found by prob. 1. Thus, 75abc 39 measure. 34, 25bc being the greatest common 125bcx 5x a4-b a2 + b2 Alfo, the greatest 9a4b-9a3b2+3a2b3—3ab4_9a3+3ab2 8a2b2-10ab3+264. 8ab2b common measure being a-bxb, by prob. 1. Prob. III. To reduce an integer to the form of a fraction. E Rule Rule. Multiply the given integer by any quantity for a numerator, and fet that quantity under the product, for a denomianatory into i D Thus, a, a+b= Cor. Hence, in the following operations concerning fractions, an integer may be introduced; for, by this problem, it may be reduced to the form of a fraction. The denominator of an integer is generally made I. Prob. IV. To reduce fractions with different denominators to fractions of equal value, that shall have the fame denomina tor. Rule. Multiply each numerator, Separately taken, into all the denominators but its own, and the products fhall give the new numerators. Then multiply all the denominators into one another, and the product Jhall give the common denominator. Example, a Example. Let the fractions be they are refpectively equal to adf bef bde bdf bdf bdf The reason of the operation appears from the preceding propofition; for the numerator and denominator of each fraction are multiplied by the fame quantities; and the value of the fractions therefore is the fame. Prob. V. To add and subtract fractions. Rule. Reduce them to a common denominator, then add or fubtract the numerators; and the fum or difference fet over the common denominator, is the fum or remainder required. From the nature of divifion it is evident, that, when feveral quantities are to be divided by the fame divifor, the fum of the quotients is the fame with the quotient of the fum of the quantities divided by that common divifor. |